Cubic or third degree equations were known to the ancient Indians and ancient Greeks since the 5th century BC, and even earlier to the ancient Egyptians, who dealt with the problem of doubling the cube, and attempted to solve it using compass and straightedge constructions.
In the early 16th century, the Italian mathematician Scipione del Ferro (1465-1526) found a method for solving a class of cubic equations, namely those of the form x3+mx=n. In fact, all cubic equations can be reduced to this form if m and n can be negative. Solution involves complex arithmetic, various conditional tests, extraction of roots and trigonometric functions. Modern general processing units can solve such equations tens of thousands of times per second. However, high speed imaging systems may generate tens of millions of pixel values per second. If their nonlinearities are sufficiently complex that a good fit requires a cubic polynomial, correction of their nonlinearities in real time by the general solution of those polynomials would require expensive, custom computational resources.
A numerical table to approximately invert a homogeneous cubic polynomial would be expected to be three dimensional to accommodate arbitrary values of the three coefficients. Such a table to correct to 1% accuracy would require many megabytes of memory, making storage and access to the table slower and more expensive. The present invention defines a method of reducing the form of an arbitrary cubic polynomial to a timing parameter and a dimensionless shape parameter which permits tabulation of all cubic polynomials in a two dimensional table and rapid correction to 1% accuracy with a table occupying tens of kilobytes. The table lookup then required to correct the nonlinearities could be implemented on hardware less capable than a general purpose CPU.
The use of imaging inspection apparatus is known, including those which utilize X-ray imaging. Such apparatus are used to inspect articles such as personal luggage of airplane travelers for such undesirable items as explosives, weapons, and drugs.
One particularly successful example of such a method is that which utilizes what is referred to in the art as “X-ray Computer Tomography” apparatus (hereinafter also referred to as XCT). XCT apparatus are in wide use in the medical field for providing medical imaging such as patient body X-rays. XCT, referred to in the medical profession simply as “CT scanning,” produces a cross sectional image from a grouping of attenuation measurements taken at different angles about an object such as a patient's chest or head, while the patient is maintained in a stationary position.
Modifications have been made to such apparatus to make them adaptable to taking images for non-medical purposes. In order to make such apparatus capable of even higher speed scanning than provided by conventional stationary apparatus, such as that useful for scanning the luggage of large numbers of travelers in a relatively shorter time period, further modifications have been made.
One such apparatus is described in U.S. Pat. No. 6,236,709, for CONTINUOUS HIGH SPEED TOMOGRAPHIC IMAGING SYSTEM AND METHOD issued to Perry et al. on May 22, 2001, in which a continuous, XCT imaging system includes a conveyor which moves a closed package along the conveyor past three spaced sensing stations. At each sensing station a plurality of X-ray sources each emit a fan beam in the same scan plane which passes through the package to a plurality of detectors opposite the X-ray sources. One scan is a vertical perpendicular scan plane relative to the direction of travel of the package along the conveyor belt and the remaining two scan planes are horizontal scan planes at right angles and transverse to the direction of travel. One horizontal scan plane is a left to right scan plane while the remaining scan plane is a right to left scan plane. Each detector provides multiple energy outputs for the same data point in a scan slice, and the detector outputs are stored until all three sensing stations have scanned the same cross sectional view of the package in three directions. Scans are sequentially taken as the package moves continuously through the sensing stations and scanned data corresponding to cross sectional views of the package is accumulated. The stored data is calibrated and normalized and then used in a Computer Tomographic algebraic reconstruction technique, where the density of a reconstructed object is determined by the attenuation which it causes in the scanning X-rays. The atomic number of the object is determined from the multiple energy scan output. In a classifier, the density and atomic number are compared to a table containing density and atomic number identification values for specific objects to be located.
Other examples of various scanning apparatus systems, including those with and without conveyors, are shown and described in the following U.S. patents.
U.S. Pat. No. 6,018,562, for APPARATUS AND METHOD FOR AUTOMATIC RECOGNITION OF CONCEALED OBJECTS USING MULTIPLE ENERGY COMPUTED TOMOGRAPHY issued to Willson on Jan. 25, 2000, describes an apparatus for automatic recognition and identification of concealed objects and features thereof, such as contraband in baggage or defects in articles of manufacture. The apparatus uses multiple energy X-ray scanning to identify targets with a spectral response corresponding to a known response of targets of interest. Detection sensitivity for both automatic detection and manual inspection are improved through the multiple-energy, multi-spectral technique. Multi-channel processing is used to achieve high throughput capability. Target identification may be verified through further analysis of such attributes as shape, texture, and context of the scan data. The apparatus uses a statistical analysis to predict the confidence level of a particular target identification. A radiograph, CT image, or both may be reconstructed and displayed on a computer monitor for visual analysis by the apparatus operator. Finally, the apparatus may receive and store input from the operator for use in subsequent target identification.
U.S. Pat. No. 5,991,358, for DATA ACQUISITION SYSTEM FOR GENERATING ACCURATE PROJECTION DATA IN A CT SCANNER issued to Dolazza et al. on Nov. 23, 1999, describes a data acquisition system for use in a CT scanner which consists of an analog-to-digital converter for generating digital signals in response to analog signals representative of projection data taken at a relatively constant sampling rate. The apparatus also uses an interpolation filter for generating projection data for a plurality of predetermined projection angles as a function of the digital signals irrespective of the sampling rate. This patent references a known system which includes an array of individual detectors disposed as a single row in the shape of an arc of a circle having a center of curvature at a certain point, referred to as the “focal spot,” where the radiation emanates from the X-ray source.
The X-ray source and the array of detectors in this known system are positioned so that the X-ray paths between the source and each of the detectors all lie in the same plane (hereinafter the “rotation plane” or “scanning plane”) which is normal to the rotation axis of the disk. Since the X-ray paths originate from what is substantially a point source and extend at different angles to the detectors, the X-ray paths form a “fan beam.” The X-rays incident on a single detector at a measuring interval during a scan are commonly referred to as a “ray,” and each detector generates an analog output signal indicative of the intensity of its corresponding ray. Since each ray is partially attenuated by all the mass in its path, the analog output signal generated by each detector is representative of an integral of the density of all the mass disposed between that detector and the X-ray source (i.e., the density of the mass lying in the detector's corresponding ray path) for that measuring interval.
U.S. Pat. No. 5,524,133, for MATERIAL IDENTIFICATION USING X-RAYS issued to Neale et al. on Jun. 4, 1996, describes an X-ray analysis device for determining the mean atomic number of a material mass by locating a broad band X-ray source on one side of a testing station and on the other, a detector, comprising a target having X-ray detectors positioned adjacent thereto. One of the detectors is positioned and adapted to receive X-rays scattered by the detector target in a generally rearward direction and the other detector is positioned and adapted to detect forwardly propagating X-rays scattered off axis typically by more than 30 degrees, due to so-called “Compton scatter.” Each of the X-ray detectors provides signals proportional to the number of X-ray photons incident thereon. The apparatus further includes means responsive to the two detector outputs which form a ratio of the number of photons detected by the two detectors and forms a numerical value thereof. A look-up table containing mean atomic numbers for given numerical ratios for different materials is used, as is a means for determining from the look-up table the atomic number corresponding to the numerical ratio obtained from the outputs of the two detectors. The atomic number is provided as an output signal.
U.S. Pat. No. 7,177,391, for IMAGING INSPECTION APPARATUS issued to Chapin et al on Feb. 13, 2007, describes an imaging inspection apparatus that utilizes a plurality of individual imaging inspection devices (e.g., X-ray Computer Tomography scanning devices) positioned on a frame for directing beams onto articles having objects therein to detect the objects based on established criteria. The apparatus utilizes a conveyor which is not physically coupled to the frame having the imaging inspection devices to pass the articles along a path of travel to an inspection location within the apparatus, whereupon the inspection devices direct beams onto the article and the beams are detected and output signals provided to a processing and analysis assembly which analyzes the signals and identifies certain objects which meet the criteria.
The above patents are incorporated herein by reference.